I know that having a child abroad, can be scary but if I am honest, I felt more relaxed about having a baby abroad than having a baby back in my own sad country, the United Kingdom. In The United Kingdom, our national hospital service, (that I once worked for, officially known as the "National Health Service", NHS), is in a state of sad decay and having a baby (or in fact any treatment) in one of our state hospitals, is a hit and miss affair, based totally on luck, state funding and the free market have done their best to destroy these once great institutions. If you have insurance, are expecting a baby and live abroad, with a few despot, flea ridden country exceptions, you should get the best treatment and the best service at any private hospital.
The last hurdle to having a baby in Moscow, is registering the birth. This is done at a place called ZAGS at Profsoyuznaya Ulitsa, This place has driven many a foreigner insane and changed their hair to white with the stress and anger caused by bureaucracy. The paperwork required to get the birth recognized, could have cost the lives of a small forest in paper. We have yet to walk this path of fire however, on very reliable information, I can tell you that to register a birth, you will need an original apostilled marriage certificate from your county of marriage. Both passports must be translated by an official translation company in Moscow. You will also need the "spravka" paper from the hospital that details your babies existence, they keep this paper, which does not help your bureaucratic quest, so you will need to make a photocopy of it to take with you to the ZAGS office, as without it you can't get a birth certificate for your baby. If you want a copy of the birth certificate, you must pay 220 Rubles in cash, in the exact money, they don't give change.
I almost forgot to tell you, that you will need to speak Russian or take a good friend to help you. If you manage to do all of this, give yourself a pat on the back, drink a case of the finest Vodka and give one bottle to your friend, your baby is now another happy member of this crazy world we all live in and has an official, beautiful name to prove it.
Good luck, to all new mums and dads around the world. Treasure your children and be strict but fair, love them, protect them and educate them.
Note: The above prices and procedures may have changed since dated.